Context:
If there was a period of time (like a year), which was broken into even pieces (like months) and there was a growth rate (such as 3% annually), a starting price (like $100), then it would be easy to figure an ending price after a year. It would be described as below, and would be solved as a time value of money problem.
T: time
T/t: time increment, where there are some number of time increments (like 12 months)
R: annual growth rate (always a positive value)
P(t=0): beginning price
P(t=T): ending price
Question Setup:
If data did not exist at regular intervals (for instance 2 months and 5 months only on a one year time frame), and these points in time were not known in advance (so that it had to be solved symbolically), like:
T: time
t(1): 1st time increment
t(2): 2nd time increment
...
t(n): nth time increment
Where the period of time represented by t(1) = date of t(1) - now, t(2) = date of t(2) - date of t(1), and so forth.
Likewise, the annual growth rate differed at each point in time for which there was data, as below:
R: annual growth rate (always a positive value)
At t(1) growth annual rate is r(1) ... and so forth.
Can P(T) (ending price) be solved for symbolically? What about any corresponding 'P' at any of the point in time where there is a known 't' and a known 'r'?
Would it be possible to use a corrective factor (a coefficient in front of the growth rate 'r') to correct for the varying length of time between any two data points (e.g. t(3) - t(2))?
If so, is it possible to symbolically estimate the amount of error that introduces to the ending value of P(t) or the other 'P's?
If we assumed to know 'r' at any point in time (with a fixed value, e.g. 3% annually, such that a fraction of the total amount of time (t(5) - t(4))/t(T) * 3% would give the applicable compounding rate (some value less than 3%), could a numerical percent error estimate be made for 'P' at any point in time?
As you can likely see, I've spent a lot of time trying to figure this one out. I tried monte-carlo simulation with a few known data sets to see if I could create expected error values for a range of growth rates, but I don't think that approach is best. I think the symbolic solution is what I need. Help is very much appreciated!
William